We propose to study and develop for practical use a new class of closed sequential tests of hypotheses which has recently been proposed by the principle investigator. Our research will have two goals: (1) to extend the theory to diverse distributions of data; and (2) to compare numerically the expected sample sizes and operating characteristics of the new tests with those several existing tests--for example, tests of fixed sample size, the sequential probability ratio test, and tests whose continuation regions are wedge-shaped. We expect our results to be especially pertinent to such statistical questions in medical research as testing hypotheses about the levels of radiation found in individuals over time, testing hypotheses about the time until recovery from an operation or onset of a disease, and testing hypotheses about the effectiveness of drugs and vaccines.